box% sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 9.2, Release Date: 2020-10-24                     │
│ Using Python 3.8.6. Type "help()" for help.                        │
└────────────────────────────────────────────────────────────────────┘
sage: R.<x1,x2,x3,s1,s2,s3> = PolynomialRing(QQ, order='lex')
sage: G = ideal(-s1+x1+x2+x3,-s2+x1*x2+x1*x3+x2*x3,-s3+x1*x2*x3).groebner_basis()
sage: G
[x1 + x2 + x3 - s1, x2^2 + x2*x3 - x2*s1 + x3^2 - x3*s1 + s2, x3^3 - x3^2*s1 + x3*s2 - s3]
sage: (x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2).reduce(G)
-2*s1*s3 + s2^2
sage: -2*(x1+x2+x3)*(x1*x2*x3) + (x1*x2+x1*x3+x2*x3)^2
x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2